promath Problems and Solutions of Basic Limits: Here we will discuss various basic problems of limits with solutions. Example 1: Evaluate `lim_{x to 0} (sin x + cos x)` Solution: `lim_{x to 0} (sin x +cos x)` `=lim_{x to 0} sin x + lim_{x to 0}cos x` `=sin 0 + cos 0` `=0+1` `=1.` Example 2: Evaluate … Read more
First, recall the main formulas in the context of exponential/logarithmic limits. They will be widely used in the problems below: Formula 1. $\lim\limits_{x \to 0} \dfrac{e^x-1}{x}=1$ Formula 2. $\lim\limits_{x \to 0} \dfrac{\log_e(1+x)}{x}=1$ Formula 3. $\lim\limits_{x \to 0} \dfrac{a^x-1}{x}=\log_e a$ Exponential Logarithmic Limits: Problems and Solutions Problem 1: $\lim\limits_{x \to 0} \dfrac{e^{ax}-1}{x}$ Solution: Put $z=ax.$ Then … Read more
Limit Formulas with Their Proofs: The limit of a function is an important concept in the theory of Calculus. In this section, we will prove all the limit formulas. Formula 1: `lim_{x to 0} frac{sin x}{x}=1` Formula 2: `lim_{x to a} frac{x^n-a^n}{x-a}=na^{n-1}` Formula 3: `lim_{x to 0} frac{e^x-1}{x}=1` Formula 4: `lim_{x to 0} frac{log_e(1+x)}{x}=1` Formula 5: `lim_{x to 0} … Read more
YouTube Shorts on Mathematics As you guys know YouTube Short Videos are trending these days and many educational channels are posting short videos, we are planning to make a list of youtube short videos on topics of mathematics. This will help many YouTubers to create content on math topics. Here we go. First, we … Read more
Let us note down the list of all limit formulas. Here we will list all the important limit formulas and see how to apply such formulas in practical examples. Limit Formulas with Properties and Applications Notation and Definition: Let f(x) be a function of x. The limit of f(x) as x goes to a (x→a) … Read more
The following topics of Integral Calculus will be covered. Please click on the link provided below you wish to read. An Introduction to Integral Calculus Integration: Definition, Properties and Formulas Proofs of all integration formulas (to be updated) Integration by substitution Integration by parts (to be updated) Integration of some special functions (to be updated) Properties of definite … Read more
The following topics of Differential Calculus will be covered. Please click on the link provided below you wish to read. Derivative: Definition, Properties, and Formulas Proofs of Derivative Formulas (by definition) Some Interesting Problems & Solutions of Derivatives: 1. Derivative implies continuity but the converse is not true 2. Mod x is continuous at x=0 but not … Read more
List of all Integral formulas | Special Integration Formula | Integration formulas with examples —————————————————————- Some Basic Integration Formulas 1. $\int dx=x+c$ 2. $\int x^n dx=\frac{x^{n+1}}{n+1}+c; (n \neq -1)$ 3. $\int e^x dx=e^x+c$ 4. $\int a^x dx=\frac{a^x}{\log|a|}+c; (a>0$ but $a \neq 1)$ 5. $\int \frac{1}{x} dx=\log x+c$ 6. $\int \frac{dx}{\sqrt{1-x^2}}=\sin^{-1} x+c$ 7. $\int \frac{dx}{1+x^2}=\tan^{-1} … Read more
Here we solve few problems of definite integrals. Problem 1: By the definition of definite integrals, find `int_a^b dx`. Solution: From the definition of definite integrals, we have `int_a^b dx=lim_{h to 0} h sum_{1}^{n} 1, quad` where ` nh=b-a` `=lim_{h to 0} h (1+1+1+ cdots text{till} n text{-terms})` `=lim_{h … Read more
Here you will find out a complete list of math formulas. Calculus 1. List of Limit Formulas 2. List of Derivative Formulas 3. Properties of Derivatives 4. List of Integration Formulas